Potential inside a uniformly charged spherical shell [closed]

Question

If the magnitude of the electric field inside a uniformly charged spherical shell is zero then is how potential a non-zero constant equal to the potential of shell itself? How does a non-zero potential exist given that there is no need to do work in moving a charge in forceless field?

Answer 1

We can first determine the electric field within the shell using Gauss' law, one of Maxwell's equations. Consider a thin shell of radius $R$ which has total surface charge $Q$. For a spherical Gaussian surface $\Sigma$ within the shell, radius $r$, Gauss' law indicates that

$$ \oint_\Sigma \mathbf{E} \cdot d\mathbf{a} = \frac{Q_{\rm enc}}{\epsilon_0} = 0,$$

since we know that $Q_{\rm enc}$, the charged enclosed by our Gaussian surface, is zero. It follows that if $Q_{\rm enc}$, it must be that $\mathbf{E} = \mathbf{0}.$

Since $\mathbf{E}=\mathbf{0}$, this implies that $V = \rm constant$ because of the relationship $\mathbf{E} = -\nabla V$. That is, the (vector) derivative of a constant is zero. This means that the interior is equipotential everywhere, and it takes no work to move a charge anywhere within the shell.

This is much like how it takes no work (against the gravitational field) to move an object horizontally, since there is no change in $mgh$. Every horizontal position along a certain altitude is at a gravitational equipotential.

Answer 2

The potential is defined relative to the infinity - not relative to the center of the shell.

Since there is no field inside the shell, the potential at any point inside the shell is equal to the potential on the surface of the shell, $V=\frac Q {4\pi\epsilon_0}$.

This is illustrated for a positively charged sphere on the diagram below copied from this Hyperphysics page.

The potential in the infinity is defined as zero and it increases as we move toward a positively charged sphere as a positive work would have to be done moving a positive charge against the electric field produced by the sphere.

Inside the sphere, the field is zero, therefore, no work needs to be done to move the charge inside the sphere and, therefore, the potential there does not change.